1*> \brief \b CPPT05
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*  Definition:
9*  ===========
10*
11*       SUBROUTINE CPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT,
12*                          LDXACT, FERR, BERR, RESLTS )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          UPLO
16*       INTEGER            LDB, LDX, LDXACT, N, NRHS
17*       ..
18*       .. Array Arguments ..
19*       REAL               BERR( * ), FERR( * ), RESLTS( * )
20*       COMPLEX            AP( * ), B( LDB, * ), X( LDX, * ),
21*      $                   XACT( LDXACT, * )
22*       ..
23*
24*
25*> \par Purpose:
26*  =============
27*>
28*> \verbatim
29*>
30*> CPPT05 tests the error bounds from iterative refinement for the
31*> computed solution to a system of equations A*X = B, where A is a
32*> Hermitian matrix in packed storage format.
33*>
34*> RESLTS(1) = test of the error bound
35*>           = norm(X - XACT) / ( norm(X) * FERR )
36*>
37*> A large value is returned if this ratio is not less than one.
38*>
39*> RESLTS(2) = residual from the iterative refinement routine
40*>           = the maximum of BERR / ( (n+1)*EPS + (*) ), where
41*>             (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
42*> \endverbatim
43*
44*  Arguments:
45*  ==========
46*
47*> \param[in] UPLO
48*> \verbatim
49*>          UPLO is CHARACTER*1
50*>          Specifies whether the upper or lower triangular part of the
51*>          Hermitian matrix A is stored.
52*>          = 'U':  Upper triangular
53*>          = 'L':  Lower triangular
54*> \endverbatim
55*>
56*> \param[in] N
57*> \verbatim
58*>          N is INTEGER
59*>          The number of rows of the matrices X, B, and XACT, and the
60*>          order of the matrix A.  N >= 0.
61*> \endverbatim
62*>
63*> \param[in] NRHS
64*> \verbatim
65*>          NRHS is INTEGER
66*>          The number of columns of the matrices X, B, and XACT.
67*>          NRHS >= 0.
68*> \endverbatim
69*>
70*> \param[in] AP
71*> \verbatim
72*>          AP is COMPLEX array, dimension (N*(N+1)/2)
73*>          The upper or lower triangle of the Hermitian matrix A, packed
74*>          columnwise in a linear array.  The j-th column of A is stored
75*>          in the array AP as follows:
76*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
77*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
78*> \endverbatim
79*>
80*> \param[in] B
81*> \verbatim
82*>          B is COMPLEX array, dimension (LDB,NRHS)
83*>          The right hand side vectors for the system of linear
84*>          equations.
85*> \endverbatim
86*>
87*> \param[in] LDB
88*> \verbatim
89*>          LDB is INTEGER
90*>          The leading dimension of the array B.  LDB >= max(1,N).
91*> \endverbatim
92*>
93*> \param[in] X
94*> \verbatim
95*>          X is COMPLEX array, dimension (LDX,NRHS)
96*>          The computed solution vectors.  Each vector is stored as a
97*>          column of the matrix X.
98*> \endverbatim
99*>
100*> \param[in] LDX
101*> \verbatim
102*>          LDX is INTEGER
103*>          The leading dimension of the array X.  LDX >= max(1,N).
104*> \endverbatim
105*>
106*> \param[in] XACT
107*> \verbatim
108*>          XACT is COMPLEX array, dimension (LDX,NRHS)
109*>          The exact solution vectors.  Each vector is stored as a
110*>          column of the matrix XACT.
111*> \endverbatim
112*>
113*> \param[in] LDXACT
114*> \verbatim
115*>          LDXACT is INTEGER
116*>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
117*> \endverbatim
118*>
119*> \param[in] FERR
120*> \verbatim
121*>          FERR is REAL array, dimension (NRHS)
122*>          The estimated forward error bounds for each solution vector
123*>          X.  If XTRUE is the true solution, FERR bounds the magnitude
124*>          of the largest entry in (X - XTRUE) divided by the magnitude
125*>          of the largest entry in X.
126*> \endverbatim
127*>
128*> \param[in] BERR
129*> \verbatim
130*>          BERR is REAL array, dimension (NRHS)
131*>          The componentwise relative backward error of each solution
132*>          vector (i.e., the smallest relative change in any entry of A
133*>          or B that makes X an exact solution).
134*> \endverbatim
135*>
136*> \param[out] RESLTS
137*> \verbatim
138*>          RESLTS is REAL array, dimension (2)
139*>          The maximum over the NRHS solution vectors of the ratios:
140*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
141*>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
142*> \endverbatim
143*
144*  Authors:
145*  ========
146*
147*> \author Univ. of Tennessee
148*> \author Univ. of California Berkeley
149*> \author Univ. of Colorado Denver
150*> \author NAG Ltd.
151*
152*> \ingroup complex_lin
153*
154*  =====================================================================
155      SUBROUTINE CPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT,
156     $                   LDXACT, FERR, BERR, RESLTS )
157*
158*  -- LAPACK test routine --
159*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
160*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
161*
162*     .. Scalar Arguments ..
163      CHARACTER          UPLO
164      INTEGER            LDB, LDX, LDXACT, N, NRHS
165*     ..
166*     .. Array Arguments ..
167      REAL               BERR( * ), FERR( * ), RESLTS( * )
168      COMPLEX            AP( * ), B( LDB, * ), X( LDX, * ),
169     $                   XACT( LDXACT, * )
170*     ..
171*
172*  =====================================================================
173*
174*     .. Parameters ..
175      REAL               ZERO, ONE
176      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
177*     ..
178*     .. Local Scalars ..
179      LOGICAL            UPPER
180      INTEGER            I, IMAX, J, JC, K
181      REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
182      COMPLEX            ZDUM
183*     ..
184*     .. External Functions ..
185      LOGICAL            LSAME
186      INTEGER            ICAMAX
187      REAL               SLAMCH
188      EXTERNAL           LSAME, ICAMAX, SLAMCH
189*     ..
190*     .. Intrinsic Functions ..
191      INTRINSIC          ABS, AIMAG, MAX, MIN, REAL
192*     ..
193*     .. Statement Functions ..
194      REAL               CABS1
195*     ..
196*     .. Statement Function definitions ..
197      CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
198*     ..
199*     .. Executable Statements ..
200*
201*     Quick exit if N = 0 or NRHS = 0.
202*
203      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
204         RESLTS( 1 ) = ZERO
205         RESLTS( 2 ) = ZERO
206         RETURN
207      END IF
208*
209      EPS = SLAMCH( 'Epsilon' )
210      UNFL = SLAMCH( 'Safe minimum' )
211      OVFL = ONE / UNFL
212      UPPER = LSAME( UPLO, 'U' )
213*
214*     Test 1:  Compute the maximum of
215*        norm(X - XACT) / ( norm(X) * FERR )
216*     over all the vectors X and XACT using the infinity-norm.
217*
218      ERRBND = ZERO
219      DO 30 J = 1, NRHS
220         IMAX = ICAMAX( N, X( 1, J ), 1 )
221         XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
222         DIFF = ZERO
223         DO 10 I = 1, N
224            DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
225   10    CONTINUE
226*
227         IF( XNORM.GT.ONE ) THEN
228            GO TO 20
229         ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
230            GO TO 20
231         ELSE
232            ERRBND = ONE / EPS
233            GO TO 30
234         END IF
235*
236   20    CONTINUE
237         IF( DIFF / XNORM.LE.FERR( J ) ) THEN
238            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
239         ELSE
240            ERRBND = ONE / EPS
241         END IF
242   30 CONTINUE
243      RESLTS( 1 ) = ERRBND
244*
245*     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
246*     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
247*
248      DO 90 K = 1, NRHS
249         DO 80 I = 1, N
250            TMP = CABS1( B( I, K ) )
251            IF( UPPER ) THEN
252               JC = ( ( I-1 )*I ) / 2
253               DO 40 J = 1, I - 1
254                  TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) )
255   40          CONTINUE
256               TMP = TMP + ABS( REAL( AP( JC+I ) ) )*CABS1( X( I, K ) )
257               JC = JC + I + I
258               DO 50 J = I + 1, N
259                  TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
260                  JC = JC + J
261   50          CONTINUE
262            ELSE
263               JC = I
264               DO 60 J = 1, I - 1
265                  TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
266                  JC = JC + N - J
267   60          CONTINUE
268               TMP = TMP + ABS( REAL( AP( JC ) ) )*CABS1( X( I, K ) )
269               DO 70 J = I + 1, N
270                  TMP = TMP + CABS1( AP( JC+J-I ) )*CABS1( X( J, K ) )
271   70          CONTINUE
272            END IF
273            IF( I.EQ.1 ) THEN
274               AXBI = TMP
275            ELSE
276               AXBI = MIN( AXBI, TMP )
277            END IF
278   80    CONTINUE
279         TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
280     $         MAX( AXBI, ( N+1 )*UNFL ) )
281         IF( K.EQ.1 ) THEN
282            RESLTS( 2 ) = TMP
283         ELSE
284            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
285         END IF
286   90 CONTINUE
287*
288      RETURN
289*
290*     End of CPPT05
291*
292      END
293